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These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

\input amstex
\documentstyle{era-ams}
%\NoRunningHeads
\topmatter
\rightheadtext{THE COHOMOLOGICAL EQUATION FOR AREA-PRESERVING FLOWS}
\title THE COHOMOLOGICAL EQUATION FOR AREA-PRESERVING \\
FLOWS ON COMPACT SURFACES \endtitle 
\author 
Giovanni Forni
\endauthor
\address Dipartimento di Matematica,
Universit\`a di Bologna,
Piazza di Porta S.Donato 5,
40127 Bologna Italy 
\endaddress
\curraddr
DPMMS, University of Cambridge, 16 Mill Lane,
CB2 1SB Cambridge UK
\endcurraddr
\email forni\@ dpmms.cam.ac.uk \endemail
\cyear{1995}
\cvolno{3}
\cvolyear{1995}
\cvol{1}
\communicatedby{Svetlana Katok}
\date July 19, 1995 \inRevisedForm December 15, 1995\enddate
\keywords Cohomological equation, area-preserving flows,
higher genus surfaces \endkeywords
\subjclass 58 \endsubjclass
\advance\pageno by 113
\abstract
We study the equation $Xu=f$ where $X$ belongs to a class of 
area-preserving vector fields, having saddle-type singularities, 
on a compact orientable surface $M$ of genus $g\geq 2$. For a 
``full measure" set of such vector fields we prove the existence, 
for any sufficiently smooth complex valued function $f$ in a 
finite codimensional subspace, of a finitely differentiable 
solution $u$. The loss of derivatives is finite, but the codimension 
increases as the differentiability required for the solution
increases, so that there are a countable number of necessary and
sufficient conditions which must be imposed on $f$, in addition 
to infinite differentiability, to obtain infinitely differentiable 
solutions.  This is related to the fact that the "Keane conjecture" 
(proved by several authors such as H.Masur, W.Veech, M.Rees, 
S.Kerckhoff, M.Boshernitzan), which implies for "almost all" 
$X$ the unique ergodicity of the flow generated by $X$ on the 
complement of its singularity set, does not extend to distributions. 
Indeed, our approach proves that, for ``almost all" $X$, the vector 
space of invariant distributions not supported at the singularities 
has infinite (countable) dimension, while according to the Keane 
conjecture the cone of invariant measures is generated by the 
invariant area form $\omega$.
\endabstract
\endtopmatter

\document
\head \S 1. Introduction \endhead
\noindent  In this announcement we describe results on the 
{\it cohomological }equation $\,Xu=f\,$, where $X$ is a smooth 
area-preserving vector field on a compact orientable surface $M$ 
of genus $g\geq 2$. Topological reasons force $X$ to have singularities,
which will be assumed to be of a canonical polynomial saddle type 
(not necessarily non-degenerate). The question we answer can be stated
as follows: {\it given a smooth complex valued function }$f$ 
{\it on }$M$, {\it is it possible to find a (smooth) 
solution }$u$ {\it on }$M$ {\it to the equation }$Xu=f$?
\smallskip
\noindent The study of cohomological equations is mainly motivated by the
problem of describing time-changes for flows. In fact, the time-change 
induced in the flow generated by a vector field $X$ by a positive
function $f$ is {\it trivial }if and only if the equation $Xu=f-1$ has
(smooth) solutions. In this case the flow produced by the time-change
is (smoothly) conjugated to the original one \cite{K-H, \S 2.2}. 
\noindent The first observation on cohomological equations is that 
each invariant measure $\mu$ for the flow $\phi^X_t$ generated by
a vector field $X$ on a compact manifold $M$ gives a necessary condition
on $f$ for the existence of a continuous solution of the equation $\,Xu=f\,$. 
In fact, if $\mu$ is a probability measure invariant for $\phi^X_t$, 
$$\int_M u\circ\phi_t^X\,d\mu\equiv \int_M u\,d\mu \leqno (1.1)$$
which implies, by taking the derivative with respect to $t$ at $t=0$,
$\int Xu \,d\mu=0$.  
\smallskip
\noindent In the case of hyperbolic (Anosov) flows (or diffeomorphisms) 
the {\it Livshitz property }holds \cite{L-M-M}. Suppose $X$ is an
Anosov flow on a compact manifold $M$. Then, for any $f\in 
C^{\infty}(M)$ satisfying the condition
$$\int_Mf d\mu=0 \,\,\,\,\text {for all invariant measures }\,\mu
\,\text { supported on periodic orbits,}\leqno (1.2)$$
the cohomological equation $\,Xu=f\,$ has a $C^{\infty}$ solution which
is unique up to additive constants. The Livshitz property is also 
true in the H\"older classes $C^{\alpha}$ ($\alpha\in {\Bbb R}^+$). 
In fact, if $f$ is assumed to be only $C^{\alpha}$, then the cohomological 
equation is still solvable and the solution $u\in C^{\alpha'}$, for any 
$\alpha'<\alpha$ \cite{L-M-M, (2.3)}. 
These results were obtained for $0<\alpha<1$ in the pioneering work 
of Livshitz in the early seventies \cite {Lv}, while the $C^{\infty}$ 
case is due to Guillemin-Kazhdan \cite{G-K1-2} for the special case of 
geodesic flows on some negatively curved manifold and to de la 
Llave-Marko-Moriyon \cite {L-M-M} for more general Anosov flows. In the  
Anosov case the range of the Lie derivative operator on functions is 
{\it closed }in $C^{\alpha}$ ($\alpha>0$) and $C^{\infty}$ by the Livshitz
property. In fact, it is described by the conditions $(1.2)$ which are all  
produced by {\it invariant measures}. Secondly, there is essentially no loss of 
derivatives, i.e. the solution $u$ is almost as regular as the given $f$. 
However, the hyperbolic nature of the dynamical system forces the existence 
of a very large (countable) number of periodic orbits so that the codimension 
of the cohomological equation (i.e. the codimension of the Lie derivative 
operator on functions) is infinite in all the function spaces where
it holds.
\smallskip 
\noindent A different behaviour is expected for uniquely ergodic 
flows. A flow (or a homemorphism) on a compact space $M$ is 
{\it uniquely ergodic }if it has a unique ergodic invariant 
probability measure. Indeed, for the simplest and best known examples
of uniquely ergodic flows, the irrational translations on tori, 
the cohomological equation is {\it generically }solvable with codimension 
$1$ in $C^{\infty}$. Another important example of uniquely ergodic flow
is given by the horocycle flow on a surface of negative curvature. In this 
case not very much is known on the properties of the cohomological equation, 
but in the constant curvature case it is not difficult to show by $SL(2,\Bbb R)$
Fourier analysis that the codimension is infinite because of the existence
of infinitely many linearly independent {\it invariant distributions}. It is
worth examining a bit more closely the properties of the cohomological equation 
on the torus, since the results which we have obtained can be seen as a
generalization to higher genus. Furthermore the torus case will serve as a 
model. Without loss of generality one can consider the two-dimensional torus.
Let
$$X_{\alpha}:=\alpha_1 {{\partial}\over {\partial x_1}}\,+
\,\alpha_2 {{\partial}\over {\partial x_2}}\,\leqno (1.3)$$
be the generator of a translation on the two-torus $T^2$. By the 
Weyl equidistribution theorem, the corresponding flow on $T^2$
is uniquely ergodic iff the ratio $\alpha_1/\alpha_2$ is 
irrational. However, in such generality no Livshitz type
property holds: the range may not be closed even in $C^{\infty}$
or in the analytic category. In fact, after the pioneering paper
by C.L. Siegel \cite{Sg} and the later landmark papers by A.N. Kolmogorov 
\cite {Kl1-2} and J.Moser \cite {Mo} which started the so-called KAM
theory, it is well known that a number theoretic condition of 
{\it Diophantine }type must be imposed on the ratio $\alpha_1/\alpha_2$
to overcome the difficulties created by the appearance of {\it small 
divisors }in the formal series expansion of the solutions. The situation 
can be explained in this simple context. It goes as follows. 
Any $f\in C^s(T^2)$ can be expanded in Fourier series with 
respect to an orthonormal basis of eigenfunctions of the 
{\it flat }Laplacian $\,\Delta:= -(\partial^2/\partial x_1^2\,\,+
\partial^2/\partial x_2^2)\,$. It is possible to choose 
a Fourier basis of eigenfunctions of $\Delta$ consisting 
at the same time of eigenfunctions of $X_{\alpha}$. This
is due to the fact that the vector field $X_{\alpha}$
{\it commutes }with $\Delta$ and the eigenfunctions of 
any {\it elliptic }partial differential operator such as the 
Laplacian $\Delta$ are smooth. Indeed,
$$X_{\alpha}e_k= 2\pi i(k,\alpha)\,e_k \,\,,\,\,\,\text{if }\,\,
e_k=\exp 2\pi i(k_1x_1+k_2x_2)\,\,,\leqno (1.4)$$
where $\,(k,\alpha):=k_1\alpha_1+k_2\alpha_2\,$, for $k=(k_1,k_2)
\in {\Bbb Z}^2$. The cohomological equation $\,X_{\alpha}u=f\,$ can
therefore be immediately solved, and the Fourier coefficients 
$u_k$ of the solution $u$ are:
$$u_k= {{f_k}\over {2\pi i (k,\alpha)}}\,,\leqno (1.5)$$
where $f_k$ are the Fourier coefficients of the function $f$. 
It follows from the irrationality of the ratio $\alpha_1/\alpha_2$
that, if $f_0=0$, the equation is {\it formally }solvable. The
condition $f_0=0$ is not surprising since $f_0=\int_{T^2}f \omega$,
where $\omega=dx_1\wedge dx_2$ is the invariant area form which
spans the ($1$-dimensional) cone of invariant measures for $X_{\alpha}$.
However, the convergence of the formal solution requires further
conditions of Diophantine type on the ``small divisors" $(k,\alpha)$, 
$k\in {\Bbb Z}^2\setminus\{0\}$, for example the existence of constants 
$C>0$, $\gamma>1$ such that
$$|(k,\alpha)|\geq {C\over {|k|^{\gamma}}}\,\,,\,\,\,\,\text{ for all }
\,\,k\in {\Bbb Z}^2\setminus\{0\}\,.\leqno \text{(DC)}$$ 
Assuming a Diophantine condition $\text{(DC)}$ the following standard
result holds. There exists a natural number $l>1$ (which is related 
to the constant $\gamma>1$ in $\text{(DC)}$) such that, if $s\geq l$,
for any $f\in C^s(T^2)$ having zero average with respect to area-form 
$\omega$, there exists a solution $u\in C^{s-l}$ (unique 
up to additive constants) of the equation $X_{\alpha}u=f$. 
The proof of this statement depends on the fact that the 
Fourier coefficients of a $C^s$ function decay polynomially 
fast (as $|k|^{-s}$) as $|k|\to +\infty$ and are therefore able 
to compensate the effect of the small divisors (which is kept 
under control by $\text{(DC)}$) if $s$ is sufficiently large.
\smallskip
\noindent Thus, irrational flows on the torus give a very important
example of finite codimension cohomological equation, where the
only condition for solvability is the obvious necessary condition
produced by the invariant area. However, this ``improvement", with
respect to the case of hyperbolic flows, has the drawback that one
loses a finite number of derivatives in the solution. In fact, the
phenomenon of {\it finite loss of differentiability }is typical of
``small divisors" problem. Solvability properties (with estimates) for 
the cohomological equation in the case of irrational translations 
are the cornerstone of the classical KAM theory, which leads 
to finite codimension smooth stability (with respect to smooth
conjugacies) of such flows under ``small" perturbations and to 
the famous results concerning the persistence of invariant tori in
Hamiltonian systems (the reader can consult \cite {Bs} for a review
of the theory). Similarly the Livshitz property was the key step
in establishing the canonical perturbation theory for Anosov flows,
which describes a complete set of invariants for the smooth stability 
of such flows under ``small" hamiltonian perturbations (\cite {C-E-G}
in the case of the geodesic flow on surfaces of constant negative
curvature, \cite{L-M-M} for more general Anosov flows).
\smallskip
\noindent A class of recurrent flows on compact orientable surfaces 
of genus $g\geq 2$, having non degenerate saddle singularities, was 
studied in a pioneering paper by A.Katok \cite{Kt}, who proved that 
they always have at most a {\it finite }number of non-trivial invariant 
probability measures. Later the notion of a {\it measured foliation }on 
a compact orientable surface was introduced by Thurston \cite {Th} in 
his classification of diffeomorphisms of surfaces. Duality with respect 
to a fixed smooth area form gives a one-to-one correspondence between 
{\it orientable }measurable foliations and a class of area-preserving 
flows to which Katok's results apply. Since the space of measured 
foliations can be interpreted as a boundary of Teichm\"uller space 
(Thurston's boundary), Teichm\"uller theory began to play a role in 
understanding the ergodic properties of such flows and related 
dynamical systems (such as interval exchange transformations and 
rational polygonal billiards). The approach based on Thurston`s theory 
(described in detail in \cite{F-L-P}) and Teichm\"uller theory led in 
the eighties to a major breakthrough. The work of several authors such 
as H.Masur, W.Veech, S.Kerckhoff, M.Rees, J.Smillie, M.Boshernitzan settled  
the problem of the generic (in the sense of measure) {\it unique ergodicity }of 
such systems, known (for interval exchange transformations) as the 
{\it Keane conjecture} \cite {Kn1-2}. Masur \cite {Ms} and Veech 
\cite{Vc} solved the Keane conjecture affirmatively, giving independent 
proofs of unique ergodicity for ``almost all" measured foliations (or 
interval exchange transformations). Later, different proofs were given 
by Rees \cite {Rs}, Kerckhoff \cite {Kr}, Boshernitzan \cite {Bn} 
and others. The methods of Teichm\"uller theory developed by Masur were 
applied to rational polygonal billiards by Kerckhoff-Masur-Smillie 
\cite {K-M-S}. The meaning of these results for area-preserving flows 
on surfaces of higher genus (whose singularities are saddle-type) is 
the following. For a ``full measure" set of such flows there are no 
non-trivial probability invariant measures besides the normalized 
invariant area. A natural question is then to determine the properties 
of the corresponding cohomological equation. The results we have
obtained show how the presence of singularities generates a sequence 
of {\it invariant distributions }which represent additional obstructions 
to the solvability of the cohomological equation, besides the obvious one 
given by the invariant area as explained above. However, the finite 
codimension property still holds if we seek finitely differentiable
solutions. We believe 
a natural and interesting direction of research is to pursue the study of 
solvability properties of the cohomological equation for uniquely ergodic 
(non-hyperbolic) flows preserving a smooth volume form and of the related 
KAM-type stability properties. Our approach seems to indicate that 
{\it the choice of a well adapted Fourier analysis is a key step in this 
kind of problem}, due to the subtle nature of the {\it cancellations }involved 
in the existence of solutions.

\newpage
\head \S 2. The results \endhead
\noindent Let $M$ be a compact orientable surface  
of genus $g\geq 2$. Let $\omega$ be a smooth area form on $M$. Let 
${\Cal E}_{\omega}(M,\Sigma)$ be the set of smooth vector fields $X$ 
on $M$ which preserve the area (i.e. the Lie derivative ${\Cal L}_X 
\omega=0$) and have $\ell\geq 1$ hyperbolic singularities (saddles) with 
(negative) indices $(i_1,...,i_{\ell})$ (satisfying $i_1+...+i_{\ell}
=2-2g$) at $\Sigma:=\{p_1,...,p_{\ell}\}\subset M$ (modeled on the 
singularities of the $1$-forms $\hbox{Im}\,(z^{-i_k}\,dz)$). The 
standard $L^2$ Sobolev spaces on the compact manifold $M$ will 
be denoted by $H^s(M)$, $s\in {\Bbb Z}$. A natural measure class
can be introduced in a standard way on ${\Cal E}_{\omega}(M,\Sigma)$
by considering the Lebesgue measure class on the fundamental cohomology
classes associated to the vector fields as follows. The {\it fundamental
class }of a vector field $X\in {\Cal E}_{\omega}(M,\Sigma)$ is the
cohomology class of the closed $1$-form $\eta_X:=\imath_X\omega$ in
$H^1(M,\Sigma;\Bbb R)$, which is a finite dimensional vector space
carrying the standard Lebesgue measure class. The following results,
concerning the solvability of the cohomological equation for vector
fields in ${\Cal E}_{\omega}(M,\Sigma)$ are proved: 
\proclaim {Theorem A} There exists a ``full measure" set 
${\Cal F}_{\omega}(M,\Sigma)\subset {\Cal E}_{\omega}(M,\Sigma)$ 
such that, for every $X\in {\Cal F}_{\omega}(M,\Sigma)$, the 
following holds. There exists a natural number $l>0$ such that, 
if $f\in H^l(M)$ is supported in a compact set $K\subset 
M\setminus\Sigma$ and satisfies $\int_M f\,\omega=0$, 
then the differential equation $Xu=f$ has a solution 
$u\in L^2_{loc}(M\setminus\Sigma)$.\endproclaim
\proclaim {Theorem B} If $X\in {\Cal F}_{\omega}(M,\Sigma)$, then for 
any $s>l$ there exists a finite number $n_s>0$ of distributions 
on $M\setminus\Sigma$, ${\Cal D}^X_1,...,{\Cal D}^X_{n_s}\in H^{-s}_{loc}
(M\setminus\Sigma)$, such that the following holds. If $f\in H^s(M)$ 
is supported in a compact set $K\subset M\setminus\Sigma$ and satisfies
$$\int_M f\,\omega=0\,\,,\,\,\,\,{\Cal D}^X_{\ell}(f)=0\,\,,\,\,\ell=1,...,
n_s\,,$$
then the differential equation $Xu=f$ has a solution $u\in H^{s-l}(M)$.
\endproclaim
\smallskip
\noindent The above solvability results are obtained through 
"a priori" estimates in Sobolev spaces. Therefore the argument proving 
Theorem B also proves the following: there exists a constant $C^s_{X,K}>0$ 
such that the solution $u$ of the equation $Xu=f$, whose existence
is the content of Theorem B, satisfies the Sobolev estimate:
$$|\!|u-\hbox{vol}\,(M)^{-1}\int_M\,u\,\omega|\!|_{s-l}\leq C^s_{X,K}\,
|\!|f|\!|_s\,.\leqno (\ast)$$
It should also be noticed that the estimated loss of derivatives 
$l>0$ in Theorem B is uniform for all vector fields $X$ in the ``full 
measure" set ${\Cal F}_{\omega}(M,\Sigma)$ (in fact the argument shows 
that $l=7$ will do). 
Estimates such as $(\ast)$ are also a key ingredient of the Nash-Moser 
iteration scheme, which is the core of the KAM method in the 
smooth category. They are therefore of interest in the attempt of 
applying the KAM method to the smooth conjugacy problem for the orbit
foliations associated to the class of area-preserving flows we are
considering. This program will hopefully be completed in a forthcoming
paper. 
\smallskip 
\noindent For area-preserving vector fields on $M$ the {\it Keane conjecture},
which establishes the unique ergodicity of ``almost all" measured foliations 
on compact higher genus surfaces, in the sense of Thurston \cite{Th}, (or, 
equivalently, of ``almost all" interval exchange transformations), can be 
stated as follows:
\proclaim {Keane conjecture} For ``almost all" $X\in{\Cal E}
_{\omega}(M,\Sigma)$, the flow $\phi^X$ of $X$ is uniquely ergodic on 
$M\setminus\Sigma$, i.e. the cone of invariant measures for $\phi^X$
on $M\setminus\Sigma$ is generated by the area induced by the invariant 
$2$-form $\omega$. \endproclaim
\noindent Thus, for ``almost all" $X\in{\Cal E}_{\omega}(M,\Sigma)$, the
flow $\phi_X$ has no other invariant measures besides the delta
measures supported at its singularities and the area $\omega$. In 
particular, the Keane conjecture implies that the dynamical system 
$(\phi_X,\omega)$ is {\it ergodic }or, equivalently, that the Lie 
derivative ${\Cal L}_X$ as a differential operator on $L^2(M,\omega)$ 
has trivial kernel (consisting only of constant functions), for 
``almost all"  $X\in{\Cal E}_{\omega}(M,\Sigma)$. This weaker form 
of the Keane conjecture, which is sufficient for our analysis, can 
be obtained as a direct consequence of Theorem A by standard ergodic 
theory. All the results that we prove in the paper are therefore 
{\it independent }of earlier proofs of the Keane conjecture, such as
\cite {Ms}, \cite {Vc}, \cite {Rs}, \cite {Bn}, \cite{Kr} and
others. The approach developed in this paper also shows that the Keane 
conjecture (in the form stated above) does not extend to invariant 
distributions. In fact, the following result is proved:
\proclaim {Theorem C} For ``almost all" $X\in{\Cal E}_{\omega}(M,\Sigma)$,
the vector space of invariant distributions on $M\setminus\Sigma$ (i.e.
those distributions $\Cal D$ satisfying the equation $X{\Cal D}=0$ 
on $M\setminus\Sigma$) has infinite (countable) dimension.\endproclaim
\head \S 3. A short description of the proofs \endhead
\subhead Measured foliations and quadratic differentials \endsubhead 
As mentioned, the orbit foliation of a vector field $X\in {\Cal E}_{\omega}
(M,\Sigma)$ is a {\it measured foliation }in the sense of Thurston \cite
{Th}, i.e. it has a transverse measure given by $|\eta_X|$ where $\eta_X:
=\imath_X\omega$ is the closed dual $1$-form. The closedness of $\eta_X$
is equivalent to the area preserving property by the identity:
 $$d\imath_X\omega +\imath_Xd\omega={\Cal L}_X\omega =0\,\,,\leqno (3.1)$$
where $d\omega=0$, since $\omega$ is a $2$-form. A pair of transverse
measured foliations having the same singularities induces a {\it complex
structure }on the complement of the singularity set, which can be uniquely
extended to the whole surface $M$, and defines a {\it holomorphic quadratic 
differential }on $M$ having the two given foliations as {\it horizontal}
and {\it vertical }foliations in the following sense. A (holomorphic)
quadratic differential $q$ is a holomorphic quadratic form on $M$. With 
respect to a local holomorphic coordinate $z=x_1+ix_2$, $q$ can be written
as $q=\phi(z)dz^2$, where $\phi$ is a locally defined holomorphic function.
Then, the horizontal foliation ${\Cal F}_q$ and the vertical foliation
${\Cal F}_{-q}$ associated to $q$ are defined as follows: ${\Cal F}_q$ is 
given by $\text{Im}\,q^{1/2}=0$ with transverse measure $|\text{Im}\,q^{1/2}|$; 
similarly, ${\Cal F}_{-q}$ is given by $\hbox{Re}\,q^{1/2}=0$ with 
transverse measure $|\hbox{Re}\,q^{1/2}|$. A measured foliation $\Cal F$
is said to be {\it realizable } if ${\Cal F}={\Cal F}_q$ for some
holomorphic quadratic differential $q$. A measured foliation is realizable
iff it is possible to find another measured foliation transverse to it.
By results contained in \cite {H-M}, ``almost all" measured foliations
are realizable. In fact, a measured foliation is realizable if it has
a $1$-dimensional dense leaf (hence by \cite{Kt} and references therein
all $1$-dimensional leaves are dense). 
\smallskip
\noindent A holomorphic quadratic differential $q$ vanishing at a finite set
of points $\Sigma$ induces a {\it flat structure }with cone type singularities
at $\Sigma$. There is in fact a flat metric $R_q$ canonically associated to $q$,
defined as $R_q:=|q|^{1/2}$ and in coordinates $R_q=|\phi(z)|^{1/2} |dz|$. This
metric has {\it cone-type singularities }in the sense that, at each $p\in \Sigma$,
there exists a neighborhood $U_p$ of $p$ in $M$ such that the metric can be written 
on $U_p\setminus\{p\}$, with respect to coordinates $(\rho,\theta)$, as
$$R_q=\bigl(d\rho^2+ (c\rho\,d\theta)^2\bigr)^{1/2}\,\,,\leqno (3.2)$$
where $c$ is a positive real number ($2\pi c$ is called the {\it 
cone angle } at $p$). In our case, the cone angle is always $>2\pi$ and 
depends on the order of vanishing of the quadratic differential $q$ at $p$. 
The reader can consult \cite {St} for properties of the geometry given by
the metric $R_q$. In the particular case of the two-torus there exists
a non-vanishing holomorphic quadratic differential defined by $q=dz^2$
with respect to a global holomorphic coordinate for any complex structure.
The horizontal (resp. vertical) foliations are spanned by the vector fields
$S=\partial/\partial x_1$ (resp. $T=\partial/\partial x_2$) and the
metric $R_q$ is the standard flat Riemannian metric. In analogy with the
torus case we proceed to establish the basic properties of Fourier
analysis for any holomorphic quadratic differential on a compact Riemann 
surface $M$.
\smallskip
\subhead Fourier analysis for quadratic differentials\endsubhead 
Since the metric $R_q$ is flat, there exists on $M\setminus\Sigma$ an
orthonormal frame $\{S,T\}$ of the tangent bundle $TM$ . The vector fields
$S$, $T$ are not defined at the singularity set $\Sigma$ of the metric
$R_q$ (the set $\Sigma$ corresponds to the set of zeros of the quadratic
differential $q$ and to the singularity set of its horizontal and vertical 
foliations). In fact, if written in coordinates at a singular point $p\in 
\Sigma$, their coefficients are divergent at $p$. The {\it Laplace-Beltrami 
operator }associated to the metric $R_q$ can be written as $\Delta_q:
=-(S^2+T^2)$. It is a well defined second order {\it elliptic} differential 
operator on $M\setminus\Sigma$, but (as for $S$, $T$) its expression 
in coordinates diverges at the singularities. It is natural to introduce 
adapted Sobolev spaces $H^s_q(M)$ as follows. The space $H^0_q(M)$, denoted
by $L_q^2(M)$, is simply the $L^2$ space defined with respect to the 
area-element $\omega_q$ of the metric $R_q$. The space $H^s_q(M)$ (for 
$s\in {\Bbb N}$) is defined as completion of $C^{\infty}(M)$ with respect 
to the norm
$$|u|_s:=(\sum_{i+j\leq s}| S^i T^ju|_0^2)^{1/2}\,\,. \leqno (3.3)$$
The Laplace-Beltrami operator $\Delta_q$ has a $L^2_q$ orthonormal 
basis of {\it weak }eigenfunctions. In fact, one can consider its 
{\it Dirichlet form }
$${\Cal Q}(u,v):=(Su,Sv)_q\,+\,(Tu,Tv)_q\,\,,\leqno (3.4)$$
(where $(\cdot,\cdot)_q$ denotes the $L^2_q$ inner product) and prove
the following: 
\proclaim {Theorem 3.1 (Spectral Theorem)} The hermitian form $\Cal Q$ 
on $L^2_q(M)$ has the following properties:
\roster
\item $\Cal Q$ is  positive semi-definite and the set 
$\text{EV}({\Cal Q})$ of its eigenvalues is a discrete subset of
$[0,+\infty)$;
\item Each eigenvalue has finite multiplicity, in particular 
the eigenvalue $0$ has multiplicity $1$ and the kernel of $\Cal Q$ 
consists only of constant functions; 
\item The space $L^2_q(M)$ decomposes as an orthogonal sum 
of eigenspaces. Furthermore, the eigenfunctions $\{e_k\}
_{k\in {\Bbb N}}$ are $C^{\infty}$ (real analytic) on $M$. \endroster
\endproclaim
\noindent It is also important to establish the {\it Weyl asymptotics}
for the eigenvalues of the  hermitian form $\Cal Q$. This formula gives
the information which we need on the rate of growth of the eigenvalues. 
For any $\Lambda >0$, let $\,N_q(\Lambda):=\text{card}\,\{\lambda\in 
\text{EV}({\Cal Q})\,|\,\lambda\leq \Lambda\}$, where each eigenvalue 
$\lambda$ is counted according to its multiplicity. 
\proclaim {Theorem 3.2 (Weyl asymptotics)} There exists a
constant $C>0$ such that
$$\lim\,N_q(\Lambda)/\Lambda\,\,=\,\,C\,\text{\rom vol}_q (M)\,\,,
\,\,\,\,\hbox{ as }\,\, \Lambda\to +\infty\,\,.$$ \endproclaim
\noindent Theorem 3.1 and 3.2 can be proved by standard methods
(the reader can consult \cite{Ch}). Theorem 3.1 was proved by
J.Cheeger in \cite {Cg} for any compact Riemann manifold with 
cone-type singularities. Theorem 3.2 can also be deduced in the 
general case by Cheeger methods. Theorem 3.2 says that the growth 
of the eigenvalues is linear in $k\in {\Bbb N}$. As a simple 
consequence we obtain that the Fourier coefficients $f_k:=(f,e_k)_q$ 
of a function $f\in H^s_q(M)$ decay polynomially (as $k^{-s}$) as 
in the torus case. However, there is an obstruction (and indeed it 
must be so) to carrying over the argument sketched above for the case of 
irrational translations on the torus: although the vector fields $S$ and 
$T$ commute (hence they commute with the Laplacian $\Delta_q$), the 
eigenfunctions $e_k\in H^1_q(M)$ but in general they {\it do not 
belong }to $H^2_q(M)$. Therefore, they {\it are not }eigenfunctions 
of the vector fields $S$, $T$ or their linear combinations. Thus, a 
different strategy must be found.
\smallskip
\subhead A unitary operator\endsubhead The circle group acts on the
space of (holomorphic) quadratic differentials as follows: $(\theta,q)
\to q_{\theta}:=(e^{-i\theta})^2 q$, for all $\theta\in S^1$. The
horizontal foliation ${\Cal F}_{\theta}$ of the quadratic differential 
$q_{\theta}$ (defined by the closed $1$-form $\text{Im} (q_{\theta})^{1/2}$)
is spanned by a vector field $S_{\theta}$, unitary with respect to the
metric $R_q$, which can be obtained through a rotation by the angle 
$\theta$ from the vector field $S$ spanning the horizontal foliation 
of $q$. It is essentially an application of the Fubini theorem to show 
that all statements concerning a ``full measure" class of area-preserving 
vector fields can be reduced to the corresponding statements about 
the family $S_{\theta}$ for ``almost all" $\theta\in S^1$ (with respect
to the $1$-dimensional Lebesgue measure). In terms of the orthonormal 
frame $\{S,T\}$ introduced earlier:
$$S_{\theta}:=\{e^{-i\theta}(S+i\,T)+e^{i\theta}(S-i\,T)\}/2\,\,. 
\leqno (3.5)$$
One is led therefore to the study of the simpler {\it Cauchy-Riemann}
operators $S\pm iT$ (these are {\it elliptic }first order differential
operators), for which the following properties can be proved:
\proclaim {Proposition 3.3} The Cauchy-Riemann operators $S\pm i\,T$
are closable operators on $L^2_q(M)$, whose closures \rom(denoted by the
same symbols\rom) have the following properties. Let ${\Cal M}_{\Sigma}$ 
(resp. ${\overline {\Cal M}}_{\Sigma}$) be the (finite dimensional) vector 
spaces of {\it meromorphic }(resp. {\it anti-meromorphic }) $L^2_q$ functions. 
\roster
\item $D(S\pm i\,T)=H^1_q(M)$ and $N(S\pm i\,T)={\Bbb C}$.
\item $R^{+}:=\hbox{Ran}\,(S+ i\,T)={\overline {\Cal M}}_{\Sigma}
^{\perp}\,\,$ and $\,\,R^{-}:=\hbox{Ran}\,(S- i\,T)={\Cal M}_{\Sigma}
^{\perp}$.
\item $S\pm i \,T:(H,{\Cal Q})\to (R^{\pm},(\,\cdot\,,\,\cdot\,)_q)$ 
are unitary operators. \endroster
\endproclaim
\noindent Consequently the operator $U_q:R^-\to R^+$, defined 
as $U_q:=(S+iT)(S-iT)^{-1}$, is a partial isometry (with respect to the 
$L^2_q$ scalar product). Therefore it can be extended by any 
isometry $J:{\Cal M}_{\Sigma}\to {\overline {\Cal M}}_{\Sigma}$ 
to a unitary operator $U_J$ on $L^2_q(M)$ for which the following 
key identities hold:
$$S_{\theta}=e^{-i\theta}\Bigl(U_J+e^{2i\theta}\Bigr)(S-i\,T)
=e^{i\theta}\Bigl(U_J^{-1}+e^{-2i\theta}\Bigr)(S+i\,T)\,\,.
\leqno (3.6)$$
Estimates for the solution of the equation $S_{\theta}u=f$ in the
Sobolev spaces $H^s_q(M)$ will therefore be related to estimates for
the resolvents ${\Cal R}^{\pm}_J(z)$ of the unitary operators $U_J$
(resp. $U_J^{-1}$) at points on the circle $S^1$ (which contains the 
spectrum). It turns out that such estimates can be obtained in the
{\it weak sense} for {\it any }unitary operator, by applying {\it 
Fatou's theory }on boundary values of holomorphic functions. In fact, 
for any $u$, $v\in L^2_q(M)$, the function $z\to ({\Cal R}_J(z)u,v)_q$  
is holomorphic on the unit disk and, by the spectral theorem for unitary 
operators, it can be represented as a Cauchy integral over the spectral 
measure $d(F(t)u,v)_q$:
$$({\Cal R}_J(z)u,v)_q={1\over {2\pi}}\int_0^{2\pi}
(z-e^{i t})^{-1}\,d(F(t)u,v)_q\,\,,\,\,\,\,|z|<1\,\,,\leqno (3.7)$$
It is a classical theorem \cite {Zy, VII.9} that, for any Borel
complex measure, the corresponding Cauchy integral $I_{\mu}(z)$
(defined as in $(3.7)$ by replacing the spectral measure by the
measure $\mu$) has the following property:
\proclaim {Lemma 3.4} The non-tangential limit $I_{\mu}(z)\to 
I_{\mu}^{\ast}(\theta)$ as $z\to e^{i\theta}$ exists almost 
everywhere with respect to the Lebesgue $1$-dimensional measure 
$\Cal L$ on $S^1$. Furthermore, there exists a constant $C>0$ 
such that the following estimates hold:
$${\Cal L}\{\theta\in S^1\,|\,\,|I_{\mu}^{\ast}(\theta)|>
\lambda \}\leq {C \over {\lambda}}\,|\!|\mu|\!|\,\,,$$
where $|\!|\mu|\!|$ denotes the total mass of the measure $\mu$. 
\endproclaim
\noindent (In fact, we need a refined version of this, whose
proof can be obtained by applying results from \cite {Rd}, 
\cite {Sn} and \cite {S-W}). 
\smallskip 
\subhead Sobolev estimates and existence of invariant distributions
\endsubhead The refined version of Lemma 3.4 can be applied to find
a {\it distributional }solution of the equation $S_{\theta}u=f$ for any
given $f\in L^2_q(M)$ having zero average and for a full measure
set of $\theta\in S^1$ (which will depend on the function $f$
considered). This is obtained essentially by proving certain
estimates in the Sobolev spaces $H^s_q(M)$. The dependence on $f$
can be eliminated by Fourier series decomposition. Here the information
on the decay rate of the Fourier coefficients of $H^s_q$ functions
plays a crucial role (thus motivating the Fourier analysis of Theorems
3.1 and 3.2). The solution is then regularized through a procedure
based on properties of Cauchy-Riemann operators. Finally, the existence 
of non-trivial distributions is proved by the following idea.
By the Riemann-Roch theorem, the dimension of the space ${\Cal M}
_{\Sigma}$ of $L^2_q$ meromorphic function has dimension equal to
the genus $g\geq 2$. Hence, there always exists a meromorphic
function $\Phi$ having zero average. By the previous construction
the equation $S_{\theta}U=\Phi$ has a distributional solution for 
almost all $\theta\in S^1$. By the commutation property ${\Cal D}
:=(S+iT)U$ is an invariant distribution, i.e. $S_{\theta}{\Cal D}=0$ 
in distributional sense.
\smallskip
\subhead Aknowledgments\endsubhead I am  grateful to Prof. J.N.Mather
who suggested to me the study of smooth properties of area-preserving 
flows on higher genus surfaces. 
\Refs
\widestnumber\key {L-M-M}
\ref\key Bn \by M.Boshernitzan
\paper A condition for minimal interval
exchange maps to be uniquely ergodic
\jour  Duke Math. J.Ast\'erisque \vol 133-134
\yr1986 
\endref
\ref\key Bs \by J.-B. Bost
\paper Tores invariantes des syst\`emes dynamiques hamiltoniens
\paperinfo S\'em. Bourbaki no. 639\jour  Duke Math. J. \vol 52
\yr1985 \pages 723-752 
\endref
\ref\key Ch \by I.Chavel\book Eigenvalues in Riemannian
Geometry \publ Academic Press\yr 1984
\endref
\ref\key Cg \by J.Cheeger\paper On the spectral geometry of spaces 
with cone-like singularities\jour Proc. Nat. Acad. Sci. U.S.A. \vol 76
\yr 1979\pages 2103-2106
\endref
\ref\key C-E-G \by P.Collet-H.Epstein-G.Gallavotti\paper Perturbations
of geodesic flows on surfaces of constant negative curvature and their
mixing properties\jour Comm. Math. Phys. \vol 95\yr1984\pages  61-112
\endref
\ref\key F-L-P \by A.Fathi-F.Laudenbach-V.Po\'enaru\paper Travaux de
Thurston sur les surfaces\jour Asterisque \vol 66-67
\yr 1979
\endref
\ref\key G-K1 \by V.Guillemin-D.Kazhdan \paper On the cohomology of certain 
dynamical systems \jour Topology \vol 19 \yr 1980 \pages 291-299
\endref
\ref\key G-K2 \bysame\paper Some inverse spectral results for negatively 
curved $n$-manifolds \jour Proc. Symp. Pure Math. Amer. Math. Soc.
 \vol 36\yr 1980\pages 153-180
\endref 
\ref\key H-M \by J.Hubbard-H.Masur\paper Quadratic differentials
and foliations\jour Acta Math. \vol 142\yr 1979\pages 221-274
\endref
\ref\key Kt \by A.Katok\paper Invariant measures of flows
on oriented surfaces\jour Dokl. Akad. Nauk. SSSR \vol 211\yr
1973 \pages 775-778 \transl\nofrills English transl. in \jour Sov. Math. 
Dokl.\vol 14\yr 1973\pages 1104-1108
\endref
\ref\key K-H \by A.Katok-B.Hasselblatt\book Introduction to the
Modern Theory of Dynamical Systems\publ Cambridge University Press
\yr 1995
\endref
\ref\key Kn1 \by M.Keane\paper Interval exchange transformations\jour 
Math. Z. \vol 141\yr 1975\pages  25-31
\endref
\ref\key Kn2 \bysame\paper Non-ergodic interval exchange 
transformations\jour Israel J. Math. \vol 2\yr 1977\pages 188-196
\endref
\ref\key Kr \by S.Kerckhoff\paper Simplicial systems for interval
exchange maps and measured foliations\jour Ergod. Th. Dynam. 
Sys. \vol 5\yr 1985\pages 257-271
\endref
\ref\key K-M-S \by S.Kerckhoff-H.Masur-J.Smillie\paper Ergodicity of
Billiard Flows and Quadratic Differentials\jour Ann. of Math. 
\vol 124\yr 1986\pages 293-311
\endref
\ref\key Kl1 \by A.N. Kolmogorov\paper On dynamical systems with
integral invariant on the torus\jour Dokl. Akad. Nauk. SSSR
\vol 93 \issue 5\yr1953\pages  763-766\lang Russian
\endref
\ref\key Kl2 \by A.N. Kolmogorov\paper On conservation of conditionally
periodic motions for a small change in Hamilton's function\jour Dokl.
Akad. Nauk. SSSR\vol 98 \yr 1954\pages 527-530 \lang Russian
\endref
\ref\key Lv\by A.Livshitz\paper Homology properties of Y-systems
\jour Math. Notes USSR Acad. Sci. \vol 10\yr 1971\pages 758-763
\endref
\ref\key L-M-M \by R. de la Llave-J.Marko-R.Moriyon\paper Canonical 
perturbation theory of Anosov systems and regularity results for
the Livsic cohomology equation\jour Ann. of Math. \vol 123\yr 1986
\pages 537-611
\endref
\ref\key Ms \by H.Masur\paper Interval exchange transformations
and measured foliations\jour Ann. of Math. \vol 115\yr 1982
\pages 168-200
\endref
\ref\key Mo \by J.Moser\paper On invariant curves of area-preserving
mappings of an annulus\jour Nachrichten der Akademie der Wissenschaften
G\"ottingen, Phys. Kl. IIa \vol 1\yr 1962\pages 1-20
\endref
\ref\key Rs \by M.Rees\paper An alternative approach to the ergodic
theory of measured foliations on surfaces\jour Ergod. Th. Dynam. 
Sys.\vol 1\yr1981\pages  461-488
\endref
\ref\key Rd \by W.Rudin\book  Real \& Complex Analysis \publ
 McGraw-Hill Book Company\publaddr  New York\yr 1987 
\endref
\ref\key Sg \by C.L. Siegel\paper Iterations of analytic functions
\jour Ann. of Math. \vol 43\yr1942\pages  807-812
\endref
\ref\key Sn \by E.M.Stein\book Singular Integrals and Differentiability
Properties of Functions\publ Princeton Univ. Press\publaddr
 Princeton\yr 1970
\endref
\ref\key S-W \by E.M.Stein-G.Weiss\book Introduction to Fourier
Analysis on Euclidean Spaces\publ Princeton Univ. Press
\publaddr Princeton\yr 1971
\endref 
\ref\key St \by K.Strebel\book Quadratic Differentials\publ 
Springer-Verlag\publaddr Berlin\yr 1984
\endref
\ref\key Th \by W.Thurston\paper On the geometry and dynamics of
diffeomorphisms of surfaces\jour Bull. Amer. Math. Soc.
\vol19 \yr 1988\pages 417-431
\endref       
\ref\key Vc \by W.Veech \paper Gauss measures for transformations on the
space of interval exchange maps\jour Ann. of Math. \vol 115
\yr 1982\pages 201-242
\endref
\ref\key Zy \by A.Zygmund\book Trigonometric Series \publ
Cambridge Univ. Press \publaddr Cambridge\yr 1959 
\finalinfo (reprinted 1990)
\endref
\endRefs
\enddocument